Semi-positone Sturm-Liouville Differential Systems on Unbounded Intervals
Main Article Content
Abstract
This work is devoted to proving existence of nontrivial positivesolutions for a system of n second-order dierential equations subject to integral boundary conditions of Riemann-Stieltjes type and posed on the positive half-line. The novelty of the results is that the nonlinearity involved in the system is sign-changing and depends on the solution and on its derivative. Existence, multiplicity, and nonexistence results of nontrivial positive solutions are obtained using some xed point theorems on suitable cones of a weighted Banach space. A numerical example is included to illustrate the applicability of our results.
Article Details
How to Cite
Djebali, S., & Mebarki, K.
(2016).
Semi-positone Sturm-Liouville Differential Systems on Unbounded Intervals.
Acta Mathematica Universitatis Comenianae, 85(2), 231-259.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/240/396
Issue
Section
Articles
References
[1] R. Aris, Introduction to the Analysis of Chemical Reactors, PrenticeHall, Inc., Englewood Cliffs, New Jersey, (1965), xi and 337 pages.
[2] R.P. Agarwal, S.R. Grace, and D. O'Regan, Existence of positive solutions to semipositone Fredholm integral equations, Funkcial. Ekvac. 45 (2002), no. 2, 223-235
[3] R.P. Agarwal and D. O'Regan, A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear Anal. 36 (1999), no. 5, Ser. B: Real World Appl., 615-622
[4] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Mathematics in Science and Engineering, Vol. 104, Academic Press, New York-London, 1973, ix+238 pp.
[5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
[6] S. Djebali and K. Mebarki, Multiple unbounded positive solutions for three-point bvps with sign-changing nonlinearities on the positive half-line, Acta Appl. Math. 109 (2010), no. 2, 361-388.
[7] S. Djebali and K. Mebarki, System of singular second-order differential equations with integral condition on the positive half-line, Electron. J. Qual. Theory Differ. Equ. 2013, no. 50, 42 pp.
[8] H. Feng and D. Bai, Existence of positive solutions for semipositone multi-point boundary value problems, Math. Comput. Modelling 54 (2011), no. 9-10, 2287-2292.
[9] C.S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Diff. Equ. 20 (2015) 117-142.
[10] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5. Academic Press, Inc., Boston, MA, 1988, viii+275 pp.
[11] J. Jiang, L. Liu, and Y. Wu, Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions, Abstr. Appl. Anal. 2012, Art. ID 696283, 21 pp.
[12] M.A. Krasnosel'ski, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron P. Noordhoff Ltd. Groningen, 1964, 381 pp.
[13] M.A. Krasnosel'ski, Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 196,4 xi + 395 pp.
[14] K.Q. Lan, Multiple positive solutions of semi-positone Sturm-Liouville boundary value problems, Bull. London Math. Soc. 38 (2006), no. 2, 283-293.
[15] R.W. Leggett and L.R. Williams, Multiple positive fixed points of nonlinear operators on ordred Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673-688.
[16] J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 1-2, 45-67.
[17] J.R.L. Webb and G. Infante, Semi-positone nonlocal boundary value problems for arbitrary order, Commun. Pure Appl. Anal. 9 (2010), no. 2, 563-581.
[18] S.L. Xi, M. Jia, and H.P. Ji, Positive solutions of boundary value problem for systems of second-order differential equations with integral boundary condition on the half line, Electron. J. Qual. Theory Differ. Equ. 2009, no. 31, 13 pp.
[19] S.L. Xi, M. Jia, and H.P. Ji, Multiple positive solutions for boundary value problems of second-order differential equations system on the half line, Electron. J. Qual. Theory Differ. Equ. 2010, no. 17, 15 pp.
[20] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. I: Fixed Point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986, xxi+897 pp.
21] X.Q. Zhang, Positive solutions of singular multipoint boundary value problem for systems of nonlinear second-order differential equations on infinite intervals in Banach spaces, Bound. Value Probl. 2009, Art. ID
978605, 22 pp.
[2] R.P. Agarwal, S.R. Grace, and D. O'Regan, Existence of positive solutions to semipositone Fredholm integral equations, Funkcial. Ekvac. 45 (2002), no. 2, 223-235
[3] R.P. Agarwal and D. O'Regan, A note on existence of nonnegative solutions to singular semi-positone problems, Nonlinear Anal. 36 (1999), no. 5, Ser. B: Real World Appl., 615-622
[4] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Mathematics in Science and Engineering, Vol. 104, Academic Press, New York-London, 1973, ix+238 pp.
[5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
[6] S. Djebali and K. Mebarki, Multiple unbounded positive solutions for three-point bvps with sign-changing nonlinearities on the positive half-line, Acta Appl. Math. 109 (2010), no. 2, 361-388.
[7] S. Djebali and K. Mebarki, System of singular second-order differential equations with integral condition on the positive half-line, Electron. J. Qual. Theory Differ. Equ. 2013, no. 50, 42 pp.
[8] H. Feng and D. Bai, Existence of positive solutions for semipositone multi-point boundary value problems, Math. Comput. Modelling 54 (2011), no. 9-10, 2287-2292.
[9] C.S. Goodrich, Semipositone boundary value problems with nonlocal, nonlinear boundary conditions, Adv. Diff. Equ. 20 (2015) 117-142.
[10] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5. Academic Press, Inc., Boston, MA, 1988, viii+275 pp.
[11] J. Jiang, L. Liu, and Y. Wu, Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions, Abstr. Appl. Anal. 2012, Art. ID 696283, 21 pp.
[12] M.A. Krasnosel'ski, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron P. Noordhoff Ltd. Groningen, 1964, 381 pp.
[13] M.A. Krasnosel'ski, Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 196,4 xi + 395 pp.
[14] K.Q. Lan, Multiple positive solutions of semi-positone Sturm-Liouville boundary value problems, Bull. London Math. Soc. 38 (2006), no. 2, 283-293.
[15] R.W. Leggett and L.R. Williams, Multiple positive fixed points of nonlinear operators on ordred Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673-688.
[16] J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 1-2, 45-67.
[17] J.R.L. Webb and G. Infante, Semi-positone nonlocal boundary value problems for arbitrary order, Commun. Pure Appl. Anal. 9 (2010), no. 2, 563-581.
[18] S.L. Xi, M. Jia, and H.P. Ji, Positive solutions of boundary value problem for systems of second-order differential equations with integral boundary condition on the half line, Electron. J. Qual. Theory Differ. Equ. 2009, no. 31, 13 pp.
[19] S.L. Xi, M. Jia, and H.P. Ji, Multiple positive solutions for boundary value problems of second-order differential equations system on the half line, Electron. J. Qual. Theory Differ. Equ. 2010, no. 17, 15 pp.
[20] E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol. I: Fixed Point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986, xxi+897 pp.
21] X.Q. Zhang, Positive solutions of singular multipoint boundary value problem for systems of nonlinear second-order differential equations on infinite intervals in Banach spaces, Bound. Value Probl. 2009, Art. ID
978605, 22 pp.